Overview on S-Box Design Principles Debdeep Mukhopadhyay Assistant - - PDF document
Overview on S-Box Design Principles Debdeep Mukhopadhyay Assistant Professor Department of Computer Science and Engineering Indian Institute of Technology Kharagpur INDIA -721302 What is an S-Box? S-Boxes are Boolean mappings from {0,1}
minimal sum (XOR +) of products (AND .) form:
function
f(x1,…,xn)= a0+ a1. x1 + …+ an. xn+ a1,2.x1.x2+ …+ an-1,n.xn-1.xn+ … …+ a1,2,..,n x1.x2 ...xn
1 2 1
n
n
−
1 2 1
( ) ( ) ( )
{( 1) ,( 1) ,...,( 1) } is called sequence of
n
f f f
f
α α α
−
− − −
n
m in ( , ) w here is the set of all affine functions over
n
f g n
N d f g A
∈ Α
∴ = Σ
1
1 1 0,1,...,2
n
n n f i i i
−
− − =
HHT=rIr, where Ir is the identity matrix of dimension rxr.
belonging to {0,1}n
1 1 1 1 1
1, , 1,2,...
n n n n
H H H H n H H
− − − −
⎡ ⎤ = = = ⎢ ⎥ − ⎣ ⎦
( ) , , is the binary representation of Note that and are not sequences, but they are binary tuples of length
i i i
g x x i x n α α α =< >
( ) ( ) is balanced, is a row of the matrix A f x f x γ γ ⊕ ⊕
1 A= 0 1 1 1 1 ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
1 1 2
n n f
− −